Bootstrapping Lieb-Schultz-Mattis anomalies

We incorporate the microscopic assumptions that lead to a certain generalization of Lieb-Schultz-Mattis (LSM) theorem for one-dimensional spin chains into conformal bootstrap. Our approach accounts ``LSM anomaly'' possessed by these through combination modular bootstrap and correlator symmetry defect operators. Specifically, we obtain universal bounds on local operator content $(1+1)d$ field theories (CFTs) could describe translationally invariant lattice Hamiltonians with ${\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N}$ realized projectively at each site. assume, such models, in CFT translation is as an emanant internal ${\mathbb{Z}}_{N}$ symmetry. present operators both without refinement their global representations. Interestingly, can nontrivial charged when $N$ odd, which turns out be impossible alone. exhibit distinctive kinks, some are approximately saturated known others unexplained. discuss additional scenarios properties necessary our apply, including multicritical points between protected topological phases, where argue anomaly studied calculations should emerge.